How do you find all the asymptotes for function #f(x) = (11(x-5)^5(x+6)^5)/ (x^10+11x^5+30)#?

1 Answer
Feb 26, 2016

Answer:

#x=-root(5)6# and #x=-root(5)5# are two vertical asymptotes. Horizontal asymptote is #y=11#.

Explanation:

To find asymptotes of #f(x)=(11(x−5)^5*(x+6)^5)/(x^10+11x^5+30)#, we should factorize denominator #(x^10+11x^5+30)#

The factors of #(x^10+11x^5+30)# are #(x^5+5)# and #(x^5+6)#

As #(x+a)# is a factor of #(x^5+a^5)# (as #-a# is zero of latter)

two factors of #(x^10+11x^5+30)# are #x=-root(5)6# and #x=-root(5)5#

and hence #x=-root(5)6# and #x=-root(5)5# are two vertical asymptotes.

It is also apparent that the highest degree of numerator is #11x^10# and that of denominator is #x^10#. As their ratio is #11#

Horizontal asymptote is #y=11#.